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May  2016, 15(3): 761-794. doi: 10.3934/cpaa.2016.15.761

A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

1. 

Department of mathematics, Korea university, 1 anam-dong, sungbuk-gu, Seoul 136-701, South Korea

2. 

Department of Mathematics, Ajou University, 206 Worldcup-ro, Yeontong-gu, Suwon 443-749, South Korea

Received  March 2015 Revised  December 2015 Published  February 2015

In this article we consider parabolic systems and $L_p$ regularity of the solutions. With zero boundary condition the solutions experience bad regularity near the boundary. This article addresses a possible way of describing the regularity nature. Our space domain is a half space and we adapt an appropriate weight into our function spaces. In this weighted Sobolev space setting we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations. Using these, we prove uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
Citation: Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure and Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
References:
[1]

M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients, Comm. Partial Differential Equations, 18 (1993), 1735-1763. doi: 10.1080/03605309308820991.

[2]

Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations, 209 (2005), 229-265. doi: 10.1016/j.jde.2004.08.018.

[3]

F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. doi: 10.2307/2154379.

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24, Pittman, Boston-London-Melbourn, 1985.

[5]

R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc., 74 (2006), 717-736. doi: 10.1112/S0024610706023192.

[6]

Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361. doi: 10.1007/s11118-007-9042-8.

[7]

N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, Journal of Functional Analysis, 183 (2001), 1-41. doi: 10.1006/jfan.2000.3728.

[8]

N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights, SIAM J. Math. Anal., 32 (2001), 1117-1141. doi: 10.1137/S0036141000372039.

[9]

N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs 64 , AMS, Providence, RI, (1999), 185-242. doi: 10.1090/surv/064/05.

[10]

N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space, Comm. in PDEs, 23 (1999), 1611-1653. doi: 10.1080/03605309908821478.

[11]

N. V. Krylov, Some properties of weighted Sobolev spaces in $\bmathbb{R}^{d}_+$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 675-693.

[12]

N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Relat. Fields, 98 (1994), 389-421. doi: 10.1007/BF01192260.

[13]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Prividence, RI, 2008. doi: 10.1090/gsm/096.

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325. doi: 10.1137/S0036141097326908.

[15]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. on Math. Anal., 31 (1999), 19-33. doi: 10.1137/S0036141098338843.

[16]

Kijung Lee, On a deterministic linear partial differential system, J. Math. Anal. Appl., 353 (2009), 24-42. doi: 10.1016/j.jmaa.2008.11.059.

[17]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$, Dunod, Paris, 1968.

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods and Applications of Analysis, 1 (2000), 195-204.

[19]

V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle, Zapiski Nauchnykh Seminarov LOMI, 138 (1984), 146-180 in Russian; English translation in Journal of Soviet Math., 32 (1986), 526-546.

[20]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983. doi: 10.1007/978-3-0346-0416-1.

show all references

References:
[1]

M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients, Comm. Partial Differential Equations, 18 (1993), 1735-1763. doi: 10.1080/03605309308820991.

[2]

Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations, 209 (2005), 229-265. doi: 10.1016/j.jde.2004.08.018.

[3]

F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. doi: 10.2307/2154379.

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24, Pittman, Boston-London-Melbourn, 1985.

[5]

R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc., 74 (2006), 717-736. doi: 10.1112/S0024610706023192.

[6]

Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361. doi: 10.1007/s11118-007-9042-8.

[7]

N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, Journal of Functional Analysis, 183 (2001), 1-41. doi: 10.1006/jfan.2000.3728.

[8]

N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights, SIAM J. Math. Anal., 32 (2001), 1117-1141. doi: 10.1137/S0036141000372039.

[9]

N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs 64 , AMS, Providence, RI, (1999), 185-242. doi: 10.1090/surv/064/05.

[10]

N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space, Comm. in PDEs, 23 (1999), 1611-1653. doi: 10.1080/03605309908821478.

[11]

N. V. Krylov, Some properties of weighted Sobolev spaces in $\bmathbb{R}^{d}_+$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 675-693.

[12]

N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Relat. Fields, 98 (1994), 389-421. doi: 10.1007/BF01192260.

[13]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Prividence, RI, 2008. doi: 10.1090/gsm/096.

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325. doi: 10.1137/S0036141097326908.

[15]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. on Math. Anal., 31 (1999), 19-33. doi: 10.1137/S0036141098338843.

[16]

Kijung Lee, On a deterministic linear partial differential system, J. Math. Anal. Appl., 353 (2009), 24-42. doi: 10.1016/j.jmaa.2008.11.059.

[17]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$, Dunod, Paris, 1968.

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods and Applications of Analysis, 1 (2000), 195-204.

[19]

V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle, Zapiski Nauchnykh Seminarov LOMI, 138 (1984), 146-180 in Russian; English translation in Journal of Soviet Math., 32 (1986), 526-546.

[20]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983. doi: 10.1007/978-3-0346-0416-1.

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